What is an end-to-end AI project?
It's a pretty grey area.

I think they are usually talking about the full spectrum of the effort in taking an "AI project" all the way from the drawing board to production. This could easily be anything from a Stock Market Price Prediction model, to deploying a customer service chatbot/Virtual Assistant. I find that people who ask such questions only care about the fact that "AI" is being used to solve a problem and actually is working in the real world. 

The effort in the Model would be the standard DataScience/Machine learning process of obtaining data, feature engineering, training and refining your model and eventually deploying it somewhere where it works. It could be in a web browser, it could be in an app, an API endpoint, etc. 

For the customer service chatbot it could easily be largely a pure programming project utilising a cloud provider's resource for to conversational capabilities, but as long as it falls under the bracket of "AI" and has been completed and deployed to production/an environment where it works, its end to end. 

Disclaimer: Depending on the technical knowledge of who is asking, they might care that you have done some real machine learning work in the development process, and so the example of the bot wouldn't fly. 

Hope this helps somehow.